The dry and raspy sound a fart makes after it escapes someone's rectum, similar to a elephant honking. If the smell is similar to fermented shit, it's a wild one, and if it doesn't reek a smell, it's a zoo-kept one. Brown Elephants vary from what the individual ate, such as a bean burrito, which will give you constant reeky toots, or a diet of greens which lowers your chance of it smelling. Since the Brown Elephant spectrum is widely diverse by smell and food, there could be ones that reek from your anus, or ones audible, but no smell.
Derrick: We all heard it in class, Jacob ripped out a Wild Brown Elephant in the middle of the essay, and we all laughed our asses off.
Raul: Of course, I heard my chair vibrate as he tore his underwear from releasing that Wild Brown Elephant
When two people put their anuses together, one person defecates (preferably a solid log), while the other relaxes to receive the gift into their rectum.
Derived from the "White Elephant" (a crappy gift - pun intended), the Brown Elephant Exchange also illustrates the exchange of items from one elephant trunk to another.
This maneuver is typically only performed by experts.